This page is going to contain an introduction to topological K-theory. For the moment it is going to be a digest of Wirthmuller 12, eventually there might be more here.
under construction
What is called topological K-theory is a fundamental tool in the field of topology and homotopy theory. Topological K-theory has an elementary definition but far-reaching applications and results in diverse fields of pure mathematics, as well as in mathematical physics.
To recall, the subject of topology studies spaces with continuous functions between them: topological spaces, a fundamental concept in all of mathematics. For many applications one cares about the choice of continuous functions only up to continuous deformations, called homotopies. The study of topological spaces with continuous functions up to homotopy is called homotopy theory. Interestingly, this plays an even more fundamental role in mathematics. For some introductory exposition see at Higher Structures in Mathematics and Physics?.
In order to study topological spaces up to homotopy – then called homotopy types – a useful strategy is to assign algebraic data to them which may be transferred along continuous functions but is invariant under homotopy: homotopy invariants. This approach to homotopy theory is called algebraic topology.
A basic example of such a homotopy invariant of topological spaces is singular homology and singular cohomology. These are sequences of abelian groups which classify formal formal linear combinations of “singular chains” in a topological space, essentially a simple algebraic way to detect which kind of “holes” there are in a topological space.
But there are more interesting and richer homotopy invariants of topological spaces. In a sense the next example after the “ordinary” singular cohomology is the “generalized cohomology theory” called topological K-theory.
In topological K-theory one detects properties of topological spaces by regarding vector bundles over them. A vector bundle over a topological space $X$ is an assignment of a vector space $V_x$ (the “fiber” over $x$) to each of its points, such that these glue together to one big space $V$ over $X$ as the point $x$ varies in $X$. Hence vector bundles are to be thought of as “continuously varying families of vector spaces”, combining topology with linear algebra.
graphics grabbed from Hatcher
For example if $X$ is differentiable, then at each of its points there is a vector space of tangent vectors called the tangent space at that point. The collection of these tangent spaces forms a vector bundle called the tangent bundle. The graphics on the right shows one tangent space to the 2-sphere.
Just as two vector spaces may be combined in their direct sum, so two vector bundles may be combined by point-wise direct sum of their fibers. This makes the vector bundles over a fixed topological space form a semi-group (in fact a monoid). This may universally be turned into an abelian group, and this is the topological K-theory group of the underlying topological space. Hence
Topological K-theory studies vector bundles over topological spaces together with their formal inverses with respect to direct sum of vector bundles.
There is a more algebraic incarnation of topological K-theory: To every topological space is assigned its algebra of continuous functions with values in the complex numbers. Given a vector bundle over the space, then these continuous functions act on the space of sections of the vector bundle by their pointwise action on the vector space fibers. This makes the collection of continuous sections of a vector bundle into a module over the algebra of functions on the topological space.
Remarkably, if the topological space is compact, then this construction is an equivalence that completely reflects the topology in operator algebra. This is the statement of Gelfand duality and the Serre-Swan theorem.
Accordingly, there are two dual incarnations of K-groups,
one in terms of algebraic topology: classifying spaces, spectra and generalized (Eilenberg-Steenrod) cohomology theory,
the other in terms of operator algebra: C*-algebras, Hilbert bimodules, Fredholm operators and the category which these form, the “KK-category”.
In the seminar, we go through the basic constructions and theorem in both of these approaches in parallel.
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We now say some of this again, at a slightly more technical level.
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Recall that for $k$ a field then a $k$-vector bundle over a topological space $X$ is a map $V \to X$ whose fibers are vector spaces which vary over $X$ in a controlled way. Explicitly this means that there exits an open cover $\{U_i \to X\}$ of $X$, a natural number $n \in \mathbb{N}$ (the rank of the vector bundle) and a homeomorphism $U_i \times k^n \to V|_{U_i}$ over $U_i$ which is fiberwise a $k$-linear map.
Vector bundles are of central interest in large parts of mathematics and physics, for instance in Chern-Weil theory and cobordism theory. But the collection $Vect(X)_{/\sim}$ of isomorphism classes of vector bundles over a given space is in general hard to analyze. One reason for this is that these are classified in degree-1 nonabelian cohomology with coefficients in the (nonabelian) general linear group $GL(n,k)$. K-theory may roughly be thought of as the result of forcing vector bundles to be classified by an abelian cohomology theory.
To that end, observe that all natural operations on vector spaces generalize to vector bundles by applying them fiber-wise. Notably there is the fiberwise direct sum of vector bundles, also called the Whitney sum operation. This operation gives the set $Vect(X)_{/\sim}$ of isomorphism classes of vector bundles the structure of an semi-group (monoid) $(Vect(X)_{/\sim},\oplus)$.
Now as under direct sum, the dimension of vector spaces adds, similarly under direct sum of vector bundles their rank adds. Hence in analogy to how one passes from the additive semi-group (monoid) of natural numbers to the addtitive group of integers by adjoining formal additive inverses, so one may adjoin formal additive inverses to $(Vect(X)_{/\sim},\oplus)$. By a general prescription (“Grothendieck group”) this is achieved by first passing to the larger class of pairs $(V_+,V_-)$ of vector bundles (“virtual vector bundles”), and then quotienting out the equivalence relation given by
for all $W \in Vect(X)_{/\sim}$. The resulting set of equivalence classes is an abelian group with group operation given on representatives by
and with the inverse of $[V_+,V_-]$ given by
This abelian group obtained from $(Vect(X)_{/\sim}, \oplus)$ is denoted $K(X)$ and often called the K-theory of the space $X$. Here the letter “K” (due to Alexander Grothendieck) originates as a shorthand for the German word Klasse, referring to the above process of forming equivalence classes of (isomorphism classes) of vector bundles.
This simple construction turns out to yield remarkably useful groups of homotopy invariants. A variety of deep facts in algebraic topology have fairly elementary proofs in terms of topolgical K-theory, for instance the Hopf invariant one problem (Adams-Atiyah 66).
One defines the “higher” K-groups of a topological space to be those of its higher suspensions
The assignment $X \mapsto K^\bullet(X)$ turns out to share many properties of the assignment of ordinary cohomology groups $X \mapsto H^n(X,\mathbb{Z})$. One says that topological K-theory is a generalized (Eilenberg-Steenrod) cohomology theory. As such it is represented by a spectrum. For $k = \mathbb{C}$ this is called KU, for $k = \mathbb{R}$ this is called KO. (There is also the unification of both in KR-theory.)
One of the basic facts about topological K-theory, rather unexpected from the definition, is that these higher K-groups repeat periodically in the degree $n$. For $k = \mathbb{R}$ the periodicity is 8, for $k = \mathbb{C}$ it is 2. This is called Bott periodicity.
It turns out that an important source of virtual vector bundles representing classes in K-theory are index bundles: Given a Riemannian spin manifold $B$, then there is a vector bundle $S \to B$ called the spin bundle of $B$, which carries a differential operator, called the Dirac operator $D$. The index of a Dirac operator is the formal difference of its kernel by its cokernel $[ker D, coker D]$. Now given a continuous family $D_x$ of Dirac operators/Fredholm operators, parameterized by some topological space $X$, then these indices combine to a class in $K(X)$.
It is via this construction that topological K-theory connects to spin geometry (see e.g. Karoubi K-theory) and index theory.
As the terminology indicates, both spin geometry and Dirac operator originate in physics. Accordingly, K-theory plays a central role in various areas of mathematical physics, for instance in the theory of geometric quantization (“spin^c quantization”) in the theory of D-branes (where it models D-brane charge and RR-fields) and in the theory of Kaluza-Klein compactification via spectral triples (see below).
All these geometric constructions have an operator algebraic incarnation: by the topological Serre-Swan theorem then vector bundles of finite rank are equivalently modules over the C*-algebra of continuous functions on the base space. Using this relation one may express K-theory classes entirely operator algebraically, this is called operator K-theory. Now Dirac operators are generalized to Fredholm operators.
There are more C*-algebras than arising as algebras of functions of topological space, namely non-commutative C-algebras. One may think of these as defining non-commutative geometry, but the definition of operator K-theory immediately generalizes to this situation (see also at KK-theory).
While the C*-algebra of a Riemannian spin manifold remembers only the underlying topological space, one may algebraically encode the smooth structure and Riemannian structure by passing from Fredholm modules to “spectral triples”. This may for instance be used to algebraically encode the spin physics underlying the standard model of particle physics and operator K-theory plays a crucial role in this.
This section serves as warmup and for background. Since the topological K-theory that we are after may be seen as a higher analog of ordinary cohomology, we first recall ordinary cohomology.
This section reviews some basic notions in topology and homotopy theory. These will all serve as blueprints for corresponding notions in homological algebra.
A topological space is a set $X$ equipped with a set of subsets $U \subset X$, called open sets, which are closed under
The Cartesian space $\mathbb{R}^n$ with its standard notion of open subsets given by unions of open balls $D^n \subset \mathbb{R}^n$.
For $Y \hookrightarrow X$ an injection of sets and $\{U_i \subset X\}_{i \in I}$ a topology on $X$, the subspace topology on $Y$ is $\{U_i \cap Y \subset Y\}_{i \in I}$.
For $n \in \mathbb{N}$, the topological n-simplex is, up to homeomorphism, the topological space whose underlying set is the subset
of the Cartesian space $\mathbb{R}^{n+1}$, and whose topology is the subspace topology induces from the canonical topology in $\mathbb{R}^{n+1}$.
For $n = 0$ this is the point, $\Delta^0 = *$.
For $n = 1$ this is the standard interval object $\Delta^1 = [0,1]$.
For $n = 2$ this is the filled triangle.
For $n = 3$ this is the filled tetrahedron.
A homomorphisms between topological spaces $f : X \to Y$ is a continuous function:
a function $f:X\to Y$ of the underlying sets such that the preimage of every open set of $Y$ is an open set of $X$.
Topological spaces with continuous maps between them form the category Top.
For $n \in \mathbb{N}$, $\n \geq 1$ and $0 \leq k \leq n$, the $k$th $(n-1)$-face (inclusion) of the topological $n$-simplex, def. 3, is the subspace inclusion
induced under the coordinate presentation of def. 3, by the inclusion
which “omits” the $k$th canonical coordinate:
The inclusion
is the inclusion
of the “right” end of the standard interval. The other inclusion
is that of the “left” end $\{0\} \hookrightarrow [0,1]$.
For $n \in \mathbb{N}$ and $0 \leq k \lt n$ the $k$th degenerate $(n)$-simplex (projection) is the surjective map
induced under the barycentric coordinates of def. 3 under the surjection
which sends
For $X \in$ Top and $n \in \mathbb{N}$, a singular $n$-simplex in $X$ is a continuous map
from the topological $n$-simplex, def. 3, to $X$.
Write
for the set of singular $n$-simplices of $X$.
As $n$ varies, this forms the singular simplicial complex of $X$. This is the topic of the next section, see def. def. 18.
For $f,g : X \to Y$ two continuous functions between topological spaces, a left homotopy $\eta : f \Rightarrow g$ is a commuting diagram in Top of the form
In words this says that a homotopy between two continuous functions $f$ and $g$ is a continuous 1-parameter deformation of $f$ to $g$. That deformation parameter is the canonical coordinate along the interval $[0,1]$, hence along the “length” of the cylinder $X \times \Delta^1$.
Left homotopy is an equivalence relation on $Hom_{Top}(X,Y)$.
The fundamental invariants of a topological space in the context of homotopy theory are its homotopy groups. We first review the first homotopy group, called the fundamental group of $X$:
For $X$ a topological space and $x : * \to X$ a point. A loop in $X$ based at $x$ is a continuous function
from the topological 1-simplex, such that $\gamma(0) = \gamma(1) = x$.
A based homotopy between two loops is a homotopy
such that $\eta(0,-) = \eta(1,-) = x$.
This notion of based homotopy is an equivalence relation.
This is directly checked. It is also a special case of the general discussion at homotopy.
Given two loops $\gamma_1, \gamma_2 : \Delta^1 \to X$, define their concatenation to be the loop
Concatenation of loops respects based homotopy classes where it becomes an associative, unital binary pairing with inverses, hence the product in a group.
For $X$ a topological space and $x \in X$ a point, the set of based homotopy equivalence classes of based loops in $X$ equipped with the group structure from prop. 3 is the fundamental group or first homotopy group of $(X,x)$, denoted
The fundamental group of the point is trivial: $\pi_1(*) = *$.
The fundamental group of the circle is the group of integers $\pi_1(S^1) \simeq \mathbb{Z}$.
This construction has a fairly straightforward generalizations to “higher dimensional loops”.
Let $X$ be a topological space and $x : * \to X$ a point. For $(1 \leq n) \in \mathbb{N}$, the $n$th homotopy group $\pi_n(X,x)$ of $X$ at $x$ is the group:
whose elements are left-homotopy equivalence classes of maps $S^n \to (X,x)$ in $Top^{*/}$;
composition is given by gluing at the base point (wedge sum) of representatives.
The 0th homotopy group is taken to be the set of connected components.
For $n = 1$ this reproduces the definition of the fundamental group of def. 11.
The homotopy theory of topological spaces is all controled by the following notion. The abelianization of this notion, the notion of quasi-isomorphism discussed in def. \ref{QuasiIsos} below is central to homological algebra.
For $X, Y \in$ Top two topological spaces, a continuous function $f : X \to Y$ between them is called a weak homotopy equivalence if
$f$ induces an isomorphism of connected components
in Set;
for all points $x \in X$ and for all $(1 \leq n) \in \mathbb{N}$ $f$ induces an isomorphism on homotopy groups
in Grp.
What is called homotopy theory is effectively the study of topological spaces not up to isomorphism (here: homeomorphism), but up to weak homotopy equivalence. Similarly, we will see that homological algebra is effectively the study of chain complexes not up to isomorphism, but up to quasi-isomorphism. But this is slightly more subtle than it may seem, in parts due to the following:
The existence of a weak homotopy equivalence from $X$ to $Y$ is a reflexive and transitive relation on Top, but it is not a symmetric relation.
Reflexivity and transitivity are trivially checked. A counterexample to symmetry is the weak homotopy equivalence between the stanard circle and the pseudocircle.
But we can consider the genuine equivalence relation generated by weak homotopy equivalence:
We say two spaces $X$ and $Y$ have the same (weak) homotopy type if they are equivalent under the equivalence relation generated by weak homotopy equivalence.
Equivalently this means that $X$ and $Y$ have the same (weak) homotopy type if there exists a zigzag of weak homotopy equivalences
One can understand the homotopy type of a topological space just in terms of its homotopy groups and how they act on each other. (This data is called a Postnikov tower of $X$.) But computing and handling homotopy groups is in general hard, famously so already for the seemingly simple case of the homotopy groups of spheres. Therefore we now want to simplify the situation by passing to a “linear/abelian approximation”.
This section discusses how the “abelianization” of a topological space by singular chains gives rise to the notion of chain complexes and their homology.
Above in def. 7 we saw that to a topological space $X$ is associated a sequence of sets
of singular simplices. Since the topological $n$-simplices $\Delta^n$ from def. 3 sit inside each other by the face inclusions of def. 5
and project onto each other by the degeneracy maps, def. 6
we dually have functions
that send each singular $n$-simplex to its $k$-face and functions
that regard an $n$-simplex as beign a degenerate (“thin”) $(n+1)$-simplex. All these sets of simplices and face and degeneracy maps between them form the following structure.
A simplicial set $S \in sSet$ is
for each $n \in \mathbb{N}$ a set $S_n \in Set$ – the set of $n$-simplices;
for each injective map $\delta_i : \overline{n-1} \to \overline{n}$ of totally ordered sets $\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}$
a function $d_i : S_{n} \to S_{n-1}$ – the $i$th face map on $n$-simplices;
for each surjective map $\sigma_i : \overline{n+1} \to \bar n$ of totally ordered sets
a function $\sigma_i : S_{n} \to S_{n+1}$ – the $i$th degeneracy map on $n$-simplices;
such that these functions satisfy the simplicial identities.
The simplicial identities satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):
$d_i \circ d_j = d_{j-1} \circ d_i$ if $i \lt j$,
$s_i \circ s_j = s_j \circ s_{i-1}$ if $i \gt j$.
$d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right.$
It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of singular simplices make $(Sing X)_\bullet$ into a simplicial set. We now briefly indicate a systematic way to see this using basic category theory, but the reader already satisfied with this statement should jump ahead to the abelianization of $(Sing X)_n$ in prop. 6 below.
The simplex category $\Delta$ is the full subcategory of Cat on the free categories of the form
This is called the “simplex category” because we are to think of the object $[n]$ as being the “spine” of the $n$-simplex. For instance for $n = 2$ we think of $0 \to 1 \to 2$ as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that generate the category $[n]$, but draw also all their composites. For instance for $n = 2$ we have_
A functor
from the opposite category of the simplex category to the category Set of sets is canonically identified with a simplicial set, def. 15.
One checks by inspection that the simplicial identities characterize precisely the behaviour of the morphisms in $\Delta^{op}([n],[n+1])$ and $\Delta^{op}([n],[n-1])$.
This makes the following evident:
The topological simplices from def. 3 arrange into a cosimplicial object in Top, namely a functor
With this now the structure of a simplicial set on the singular simplices $(Sing X)_\bullet$, def. 7, is manifest: it is just the nerve of $X$ with respect to $\Delta^\bullet$, namely:
For $X$ a topological space its simplicial set of singular simplicies (often called the singular simplicial complex)
is given by composition of the functor from example 7 with the hom functor of Top:
It turns out that homotopy type of the topological space $X$ is entirely captured by its singular simplicial complex $Sing X$ (this is the content of the homotopy hypothesis-theorem).
Now we abelianize the singular simplicial complex $(Sing X)_\bullet$ in order to make it simpler and hence more tractable.
A formal linear combination of elements of a set $S \in$ Set is a function
such that only finitely many of the values $a_s \in \mathbb{Z}$ are non-zero.
Identifying an element $s \in S$ with the function $S \to \mathbb{Z}$, which sends $s$ to $1 \in \mathbb{Z}$ and all other elements to 0, this is written as
In this expression one calls $a_s \in \mathbb{Z}$ the coefficient of $s$ in the formal linear combination.
For $S \in$ Set, the group of formal linear combinations $\mathbb{Z}[S]$ is the group whose underlying set is that of formal linear combinations, def. 19, and whose group operation is the pointwise addition in $\mathbb{Z}$:
For the present purpose the following statement may be regarded as just introducing different terminology for the group of formal linear combinations:
The group $\mathbb{Z}[S]$ is the free abelian group on $S$.
For $S_\bullet$ a simplicial set, def. 15, the free abelian group $\mathbb{Z}[S_n]$ is called the group of (simplicial) $n$-chains on $S$.
For $X$ a topological space, an $n$-chain on the singular simplicial complex $Sing X$ is called a singular $n$-chain on $X$.
This construction makes the sets of simplices into abelian groups. But this allows to formally add the different face maps in the simplicial set to one single boundary map:
For $S$ a simplicial set, its alternating face map differential in degree $n$ is the linear map
defined on basis elements $\sigma \in S_n$ to be the alternating sum of the simplicial face maps:
The simplicial identity, def. 16 part (1), implies that the alternating sum boundary map of def. 23 squares to 0:
By linearity, it is sufficient to check this on a basis element $\sigma \in S_n$. There we compute as follows:
Here
the first equality is (1);
the second is (1) together with the linearity of $d$;
the third is obtained by decomposing the sum into two summands;
the fourth finally uses the simplicial identity def. 16 (1) in the first summand;
the fifth relabels the summation index $j$ by $j +1$;
the last one observes that the resulting two summands are negatives of each other.
Let $X$ be a topological space. Let $\sigma^1 : \Delta^1 \to X$ be a singular 1-simplex, regarded as a 1-chain
Then its boundary $\partial \sigma \in H_0(X)$ is
or graphically (using notation as for orientals)
In particular $\sigma$ is a 1-cycle precisely if $\sigma(0) = \sigma(1)$, hence precisely if $\sigma$ is a loop.
Let $\sigma^2 : \Delta^2 \to X$ be a singular 2-chain. The boundary is
Hence the boundary of the boundary is:
For $S$ a simplicial set, we call the collection
of abelian groups of chains $C_n(S) \coloneqq \mathbb{Z}[S_n]$, prop. 6;
and boundary homomorphisms $\partial_n : C_{n+1}(S) \to C_n(X)$, def. 23
(for all $n \in \mathbb{N}$) the alternating face map chain complex of $S$:
Specifically for $S = Sing X$ we call this the singular chain complex of $X$.
This motivates the general definition:
A chain complex of abelian groups $C_\bullet$ is a collection $\{C_n \in Ab\}_{n}$ of abelian groups together with group homomorphisms $\{\partial_n : C_{n+1} \to C_n\}$ such that $\partial \circ \partial = 0$.
We turn to this definition in more detail in the next section. The thrust of this construction lies in the fact that the chain complex $C_\bullet(Sing X)$ remembers the abelianized fundamental group of $X$, as well as aspects of the higher homotopy groups: in its chain homology.
For $C_\bullet(S)$ a chain complex as in def. 24, and for $n \in \mathbb{N}$ we say
an $n$-chain of the form $\partial \sigma \in C(S)_n$ is an $n$-boundary;
a chain $\sigma \in C_n(S)$ is an $n$-cycle if $\partial \sigma = 0$
(every 0-chain is a 0-cycle).
By linearity of $\partial$ the boundaries and cycles form abelian sub-groups of the group of chains, and we write
for the group of $n$-boundaries, and
for the group of $n$-cycles.
This means that a singular chain is a cycle if the formal linear combination of the oriented boundaries of all its constituent singular simplices sums to 0.
More generally, for $R$ any unital ring one can form the degreewise free module $R[Sing X]$ over $R$. The corresponding homology is the singular homology with coefficients in $R$, denoted $H_n(X,R)$. This generality we come to below in the next section.
For $C_\bullet(S)$ a chain complex as in def. 24 and for $n \in \mathbb{N}$, the degree-$n$ chain homology group $H_n(C(S)) \in Ab$ is the quotient group
of the $n$-cycles by the $n$-boundaries – where for $n = 0$ we declare that $\partial_{-1} \coloneqq 0$ and hence $Z_0 \coloneqq C_0$.
Specifically, the chain homology of $C_\bullet(Sing X)$ is called the singular homology of the topological space $X$.
One usually writes $H_n(X, \mathbb{Z})$ or just $H_n(X)$ for the singular homology of $X$ in degree $n$.
So $H_0(C_\bullet(S)) = C_0(S)/im(\partial_0)$.
For $X$ a topological space we have that the degree-0 singular homology
is the free abelian group on the set of connected components of $X$.
For $X$ a connected, orientable manifold of dimension $n$ we have
The precise choice of this isomorphism is a choice of orientation on $X$. With a choice of orientation, the element $1 \in \mathbb{Z}$ under this identification is called the fundamental class
of the manifold $X$.
Given a continuous map $f : X \to Y$ between topological spaces, and given $n \in \mathbb{N}$, every singular $n$-simplex $\sigma : \Delta^n \to X$ in $X$ is sent to a singular $n$-simplex
in $Y$. This is called the push-forward of $\sigma$ along $f$. Accordingly there is a push-forward map on groups of singular chains
These push-forward maps make all diagrams of the form
commute.
It is in fact evident that push-forward yields a functor of singular simplicial complexes
From this the statement follows since $\mathbb{Z}[-] : sSet \to sAb$ is a functor.
Therefore we have an “abelianized analog” of the notion of topological space:
For $C_\bullet, D_\bullet$ two chain complexes, def. 25, a homomorphism between them – called a chain map $f_\bullet : C_\bullet \to D_\bullet$ – is for each $n \in \mathbb{N}$ a homomorphism $f_n : C_n \to D_n$ of abelian groups, such that $f_n \circ \partial^C_n = \partial^D_n \circ f_{n+1}$:
Composition of such chain maps is given by degreewise composition of their components. Clearly, chain complexes with chain maps between them hence form a category – the category of chain complexes in abelian groups, – which we write
Accordingly we have:
Sending a topological space to its singular chain complex $C_\bullet(X)$, def. 24, and a continuous map to its push-forward chain map, prop. 8, constitutes a functor
from the category Top of topological spaces and continuous maps, to the category of chain complexes.
In particular for each $n \in \mathbb{N}$ singular homology extends to a functor
We close this section by stating the basic properties of singular homology, which make precise the sense in which it is an abelian approximation to the homotopy type of $X$. The proof of these statements requires some of the tools of homological algebra that we develop in the later chapters, as well as some tools in algebraic topology.
If $f : X \to Y$ is a continuous map between topological spaces which is a weak homotopy equivalence, def, 13, then the induced morphism on singular homology groups
is an isomorphism.
(A proof (via CW approximations) is spelled out for instance in (Hatcher, prop. 4.21)).
We therefore also have an “abelian analog” of weak homotopy equivalences:
For $C_\bullet, D_\bullet$ two chain complexes, a chain map $f_\bullet : C_\bullet \to D_\bullet$ is called a quasi-isomorphism if it induces isomorphisms on all homology groups:
In summary: chain homology sends weak homotopy equivalences to quasi-isomorphisms. Quasi-isomorphisms of chain complexes are the abelianized analog of weak homotopy equivalences of topological spaces.
In particular we have the analog of prop. 11:
The relation “There exists a quasi-isomorphism from $C_\bullet$ to $D_\bullet$.” is a reflexive and transitive relation, but it is not a symmetric relation.
Reflexivity and transitivity are evident. An explicit counter-example showing the non-symmetry is the chain map
from the chain complex concentrated on the morphism of multiplication by 2 on integers, to the chain complex concentrated on the cyclic group of order 2.
This clearly induces an isomorphism on all homology groups. But there is not even a non-zero chain map in the other direction, since there is no non-zero group homomorphism $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$.
Accordingly, as for homotopy types of topological spaces, in homological algebra one regards two chain complexes $C_\bullet$, $D_\bullet$ as essentially equivalent – “of the same weak homology type” – if there is a zigzag of quasi-isomorphisms
between them. This is made precise by the central notion of the derived category of chain complexes. We turn to this below in section Derived categories and derived functors.
But quasi-isomorphisms are a little coarser than weak homotopy equivalences. The singular chain functor $C_\bullet(-)$ forgets some of the information in the homotopy types of topological spaces. The following series of statements characterizes to some extent what exactly is lost when passing to singular homology, and which information is in fact retained.
First we need a comparison map:
(Hurewicz homomorphism)
For $(X,x)$ a pointed topological space, the Hurewicz homomorphism is the function
from the $k$th homotopy group of $(X,x)$ to the $k$th singular homology group defined by sending
a representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the fundamental class $[S_k] \in H_k(S^k)$, example 10.
For $X$ a topological space the Hurewicz homomorphism in degree 0 exhibits an isomorphism between the free abelian group $\mathbb{Z}[\pi_0(X)]$ on the set of path connected components of $X$ and the degree-0 singular homlogy:
Since a homotopy group in positive degree depends on the homotopy type of the connected component of the base point, while the singular homology does not depend on a basepoint, it is interesting to compare these groups only for the case that $X$ is connected.
For $X$ a path-connected topological space the Hurewicz homomorphism in degree 1
is surjective. Its kernel is the commutator subgroup of $\pi_1(X,x)$. Therefore it induces an isomorphism from the abelianization $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:
For higher connected $X$ we have the
If $X$ is (n-1)-connected for $n \geq 2$ then
is an isomorphism.
This is known as the Hurewicz theorem.
Examples
metric structure
over compact topological spaces
direct summand of trivial bundle
classifying spaces for vector bundles.
useful below for constructing the KU spectrum
We use the above fact (…) that
every real vector bundle is isomorphic to the canonical associated bundle to an O(n)-principal bundle;
every complex vector bundle is isomorphic to the canonical associated bundle to an U(n)-principal bundle.
In the following we take Top to denote compactly generated topological spaces. For these the Cartesian product $X \times (-)$ is a left adjoint and hence preserves colimits.
For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th real Stiefel manifold of $\mathbb{R}^k$ is the coset topological space.
where the action of $O(k-n)$ is via its canonical embedding $O(k-n)\hookrightarrow O(k)$.
Similarly the $n$th complex Stiefel manifold of $\mathbb{C}^k$ is
here the action of $U(k-n)$ is via its canonical embedding $U(k-n)\hookrightarrow U(k)$.
For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th real Grassmannian of $\mathbb{R}^k$ is the coset topological space.
where the action of the product group is via its canonical embedding $O(n)\times O(k-n) \hookrightarrow O(n)$ into the orthogonal group.
Similarly the $n$th complex Grassmannian of $\mathbb{C}^k$ is the coset topological space.
where the action of the product group is via its canonical embedding $U(n)\times U(k-n) \hookrightarrow U(n)$ into the unitary group.
$Gr_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n$ is real projective space of dimension $n$.
$Gr_1(\mathbb{C}^{n+1}) \simeq \mathbb{C}P^n$ is complex projective space of dimension $n$.
For all $n \leq k \in \mathbb{N}$, the canonical projection from the real Stiefel manifold (def. 32) to the Grassmannian is a $O(n)$-principal bundle
and the projection from the complex Stiefel manifold to the Grassmannian us a $U(n)$-principal bundle:
By (this cor. and this prop.).
By def. 33 there are canonical inclusions
and
for all $k \in \mathbb{N}$. The colimit (in Top, see there) over these inclusions is denoted
and
respectively.
Moreover, by def. 32 there are canonical inclusions
and
respectively, that are compatible with the $O(n)$-action and the $U(n)$-action, respectively. The colimit (in Top, see there) over these inclusions, regarded as equipped with the induced action, is denoted
and
respectively. The inclusions are in fact compatible with the bundle structure from prop. 13, so that there are induced projections
and
respectively. These are the standard models for the universal principal bundles for $O$ and $U$, respectively. The corresponding associated vector bundles
and
are the corresponding universal vector bundles.
Since the Cartesian product $O(n)\times (-)$ in compactly generated topological spaces preserves colimits, it follows that the colimiting bundle is still an $O(n)$-principal bundle
and anlogously for $E U(n)$.
As such this is the standard presentation for the $O(n)$-universal principal bundle. Its base space $B O(n)$ is the corresponding classifying space.
There are canonical inclusions
and
given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. 34 these induce maps of classifying spaces
and
There are canonical maps
and
given by sending ambient spaces and subspaces to their direct sum.
Under the colimit of def. 34 these induce maps of classifying spaces
and
The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. 33 admit the structure of CW-complexes. Moreover the canonical inclusions
and
are subcomplex incusions (hence relative cell complex inclusions).
Accordingly there is an induced CW-complex structure on the classifying spaces $B O(n)$ and $B U(n)$ (def. 34).
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
The Stiefel manifold $V_n(\mathbb{R}^k)$ from def. 32 admits the structure of a CW-complex.
e.g. (James 59, p. 3, James 76, p. 5 with p. 21, Blaszczyk 07)
(And I suppose with that cell structure the inclusions $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ are subcomplex inclusions.)
The Stiefel manifold $V_n(\mathbb{R}^k)$ (def. 32) is (k-n-1)-connected.
Consider the coset quotient projection
Since the orthogonal groups is compact (prop.) and by this corollary the projection $O(k)\to O(k)/O(k-n)$ is a Serre fibration. Therefore there is induced the long exact sequence of homotopy groups of this fiber sequence, and by this prop. it has the following form in degrees bounded by $n$:
This implies the claim. (Exactness of the sequence says that every element in $\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k))$ is in the kernel of zero, hence in the image of 0, hence is 0 itself.)
Similarly:
The complex Stiefel manifold $V_n(\mathbb{C}^k)$ (def. 32) is 2(k-n)-connected.
Consider the coset quotient projection
By prop. \ref{UnitaryGroupIsCompact} and by this corollary the projection $U(k)\to U(k)/U(k-n)$ is a Serre fibration. Therefore there is induced the long exact sequence of homotopy groups of this fiber sequence, and by prop. \ref{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:
This implies the claim.
The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. 34 is weakly contractible.
The colimiting space $E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)$ from def. 34 is weakly contractible.
The homotopy groups of the classifying spaces $B O(n)$ and $B U(n)$ (def. 34) are those of the orthogonal group $O(n)$ and of the unitary group $U(n)$, respectively, shifted up in degree: there are isomorphisms
and
(for homotopy groups based at the canonical basepoint).
Consider the sequence
from def. 34, with $O(n)$ the fiber. Since (by this prop.) the second map is a Serre fibration, this is a fiber sequence and so it induces a long exact sequence of homotopy groups of the form
Since by cor. 1 $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that
is an isomorphism.
The same kind of argument applies to the complex case.
For $n \in \mathbb{N}$ there are homotopy fiber sequences
and
exhibiting the n-sphere ($(2n+1)$-sphere) as the homotopy fiber of the canonical maps from def. 35.
This means that there is a replacement of the canonical inclusion $B O(n) \hookrightarrow B O(n+1)$ (induced via def. 34) by a Serre fibration
such that $S^n$ is the ordinary fiber of $B O(n)\to \tilde B O(n+1)$, and analogously for the complex case.
Take $\tilde B O(n) \coloneqq (E O(n+1))/O(n)$.
To see that the canonical map $B O(n)\longrightarrow (E O(n+1))/O(n)$ is a weak homotopy equivalence consider the commuting diagram
By this prop. both bottom vertical maps are Serre fibrations and so both vertical sequences are fiber sequences. By prop. 18 part of the induced morphisms of long exact sequences of homotopy groups looks like this
where the vertical and the bottom morphism are isomorphisms. Hence also the to morphisms is an isomorphism.
That $B O(n)\to \tilde B O(n+1)$ is indeed a Serre fibration follows again with this prop., which gives the fiber sequence
The claim in then follows since (this exmpl.)
The argument for the complex case is of the same form, concluding now with the identification (this exmpl.)
For $X$ a paracompact topological space, the operation of pullback of the universal principal bundle $E O(n) \to B O(n)$ from def. 34 along continuous functions $f \colon X \to B O(n)$ eastblishes a bijection
between homotopy classes of functions from $X$ to $B O(n)$ and isomorphism classes of $O(n)$-principal bundles on $X$.
A full proof is spelled out in (Hatcher, section 1.2, theorem 1.16)
Wirthmuller 12, p. 17 to p. 29
The proof of the Hopf invariant one theorem using the Adams operations in toplogical K-theory? is due to
This is reproduced for instance in
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 10.6 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Quick review includes
Klaus Wirthmüller, Vector bundles and K-theory, 2012 (pdf)